Existence theorems for quasilinear multivalued boundary value problems in R N

Abstract

Abstract. In this paper we study quasilinear second order boundary value problems with multivalued right hand side and Dirichlet boundary conditions. We prove three existence theorems. The ®rst two deal with the``convex'' and``nonconvex'' problems respectively, while the third establishes the existence of extremal solutions. For the ®rst two the proof is based on the theory of nonlinear operators of monotone type, while the proof of the third uses a ®xed point argument.1991 Mathematics Subject Classi®cation. 34A60, 34B15.1. Introduction. In this paper we study quasilinear, second order di erential inclusions in R N with Dirichlet boundary conditions. Problems of this kind for scalar ordinary di erential equations were studied by Boccardo-Drabek-GiachettiKucera [3] and Pino-Elgueta-Manasevich [14], using degree theoretic techniques. Here in addition to the vectorial and multivalued character of the problem, we also propose a di erent approach based on the theory of multivalued operators of monotone type. Moreover, in contrast to the above mentioned works, here the multivalued perturbation term F depends also on the derivative of the unknown function.After the presentation of some auxiliary results in section 2, in section 3 we prove two existence theorems. The ®rst is for the``convex'' problem (i.e. we assume that the multivalued term F tY xY y is convex valued) and the second is for thè`n onconvex'' problem (i.e. we no longer require that F tY xY y be convex valued). In section 4, we replace F tY xY y by its extreme points ext F tY xY y and look for`e xtremal solutions''. Under some stronger continuity conditions on F, we prove that such solutions exist. In contrast to section 3, our method of proof of the result in section 4, is based on a ®xed point argument which uses Schauder's ®xed point theorem. It appears that our result is the ®rst existence theorem for extremal solutions for quasilinear multivalued boundary value problems.Our results here extend the semilinear works p 2 of Frigon-Granas . The multivalued problems studied here arise naturally in many applied situations of interest, like control systems with a priori feedback, deterministic systems with uncertainties which are modelled with multifunctions and problems with discontinuous right hand side. This is the case in many problems of mathematical physics and mechanics.